write each of the following in terms of the sine of another angle between 0 degrees and 360 degrees. a) sin 60, b) sin 80, c) sin 115, d) sin 200, e) sin 300, f) 265

To express each sine function in terms of the sine of another angle between \(0^\circ\) and \(360^\circ\), we will use the properties of the sine function and the unit circle. The sine function has symmetry properties:

1. **Sine Symmetry Properties:**
   - \(\sin(180^\circ - \theta) = \sin(\theta)\)
   - \(\sin(180^\circ + \theta) = -\sin(\theta)\)
   - \(\sin(360^\circ - \theta) = -\sin(\theta)\)

Now let's apply these properties to each given angle.

### a) \(\sin(60^\circ)\)

\[
\sin(60^\circ) = \sin(60^\circ) \quad \text{(already between 0 and 360 degrees)}
\]

### b) \(\sin(80^\circ)\)

\[
\sin(80^\circ) = \sin(80^\circ) \quad \text{(already between 0 and 360 degrees)}
\]

### c) \(\sin(115^\circ)\)

\[
\sin(115^\circ) = \sin(115^\circ) \quad \text{(already between 0 and 360 degrees)}
\]

### d) \(\sin(200^\circ)\)

Using the property \(\sin(180^\circ + \theta) = -\sin(\theta)\):

\[
\sin(200^\circ) = \sin(180^\circ + 20^\circ) = -\sin(20^\circ)
\]

So, \(\sin(200^\circ) = -\sin(20^\circ)\).

### e) \(\sin(300^\circ)\)

Using the property \(\sin(360^\circ - \theta) = -\sin(\theta)\):

\[
\sin(300^\circ) = \sin(360^\circ - 60^\circ) = -\sin(60^\circ)
\]

So, \(\sin(300^\circ) = -\sin(60^\circ)\).

### f) \(\sin(265^\circ)\)

Using the property \(\sin(180^\circ + \theta) = -\sin(\theta)\):

\[
\sin(265^\circ) = \sin(180^\circ + 85^\circ) = -\sin(85^\circ)
\]

So, \(\sin(265^\circ) = -\sin(85^\circ)\).

### Summary of Results

a) \(\sin(60^\circ) = \sin(60^\circ)\)  
b) \(\sin(80^\circ) = \sin(80^\circ)\)  
c) \(\sin(115^\circ) = \sin(115^\circ)\)  
d) \(\sin(200^\circ) = -\sin(20^\circ)\)  
e) \(\sin(300^\circ) = -\sin(60^\circ)\)  
f) \(\sin(265^\circ) = -\sin(85^\circ)\)  

Would you like more details or have any questions?

### Related Questions:
1. What are the symmetry properties of cosine?
2. How do trigonometric identities work for angles greater than 360 degrees?
3. Can you express these angles in terms of the cosine function?
4. How do you convert between radians and degrees for sine and cosine functions?
5. What are the graphs of the sine function and their symmetries?

**Tip:** Remember, sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.

write each of the following in terms of the sine of another angle between 0 degrees and 360 degrees. a) sin 60, b) sin 80, c) sin 115, d) sin 200, e) sin 300, f) sin 265

Learn how to express sine functions of angles such as 60°, 80°, 115°, 200°, 300°, and 265° in terms of the sine of another angle between 0° and 360°. This solution uses trigonometric properties of symmetry and the unit circle. Suitable for students in Grades 10-12.

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